Now suppose \(Z \le \nu \) . If also event \(E\)  occurs
then it is not possible to find paths in \(S_{N+1}({\bf x})\)  both
from \(D_K({\bf x})\)  to \(J^+\) , and from \(D_K({\bf x}) \)  to \(J^-\) ,
with neither path passing through \(\cup _{\lbrace i: {\cal C}\cap Q_i \ne \varnothing \rbrace }Q_i^+\) 
(see Figure 1, and also [1]}; note
\(I \cap J^+ = I \cap J^- = \varnothing \)  since
\(\alpha _{3N}/4 < \alpha _N\) ).
Therefore we have not yet achieved event \(A^+ \cap A^-\) 
at this stage.
<FIGURE>